MATHS 361: Partial Differential Equation Tutorial 8: Fourier transforms
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MATHS 361: Partial Differential Equation
Tutorial 8: Fourier transforms
1. Use the definition of Fourier transforms to show that all even functions have a purely real Fourier transform.
2. Consider the‘sgn’function. sgn(t) = 1 if t > 0 and sgn(t) = − 1 if t < 0. Can you calculate the Fourier transform of sgn(t) directly? What limit could you take in order to calculate F(sgn(t))?
3. Use Fourier transforms to solve the partial differential equation:
utt = 4uxx ,u(x,0) = e−|x |,ut(x, 0) = 0
You may assume u → 0 at infinity for all t
4. Use Fourier transforms to solve the Partial differential equation
ut+^tux = 0,u(x,0) = H(x + 1)H(1 − x)
5. Use Fourier transforms to solve the Partial differential equation:
ut = uxx ,u(x, 0) = e −x2 = f(x)
.
6. Bonus question:
Take your answer to the previous questions, and think what happens when you run your equation backward in time. IE, what does u(x,t) looks like for t = −1, t = − 10 and so on. Do your answers always make sense? When do you run into trouble?
7. Reflections
I have the equation
uxx + uyy = 1,u(x,0) = δ(1 − x), 0 < x,y
With the assumption that u → 0 and x or y approach infinity.
(a) Suppose I told you that u(0,y) = 0. How would extend the above PDE in order to give a PDE
over −∞ < x < ∞ .
(b) Suppose I told you that ux (0,y) = 0. How would extend the above PDE in order to give a PDE
over −∞ < x < ∞ .
(c) Hard. Bonus Question. Consider attacking the rest of the tutorial first. Solve the u(0,y) = 0 case using fourier transforms. It may be useful to note that F ( sgn(t)) =
2023-06-27